We address the problem of minimizing objectives from the class of piecewise differentiable functions whose nonsmoothness can be encapsulated in the absolute value function. They possess local piecewise linear approximations with a discrepancy that can be bounded by a quadratic proximal term. This overestimating local model is continuous but generally nonconvex. It can be generated in its absnormal form by a minor extension of standard algorithmic differentiation tools. Here we demonstrate how the local model can be minimized by a bundle-type method, which benefits from the availability of additional gray-box information via the absnormal form. In the convex case our algorithm realizes the consistent steepest descent trajectory for which finite convergence was established earlier, specifically covering counterexamples where steepest descent with exact line-search famously fails. The analysis of the abs-normal representation and the design of the optimization algorithm are geared toward the general case, whereas the convergence proof so far covers only the convex case.

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Abstract. We analyze the sensitivity of dielectric waveguides with respect to design parameters such as permittivity values or simple geometric dependencies. Based on a discretization using the Finite Integration Technique the eigenvalue problem for the wave number is shown to be non-Hermitian with possibly complex solutions even in the lossless case. Nevertheless, the sensitivity can be obtained with negligible numerical effort. Numerical examples demonstrate the validity of the approach.

Nonsmoothness is a typical characteristic of numerous objective functions in optimisation that arises from applications. The standard approach in algorithmic differentiation (AD) is to consider only differentiable functions that are defined by an evaluation program. We extend this functionality by allowing also the functions abs(), min() and max() during the function evaluation yielding piecewise differential nonlinear functions. We will define an evaluation procedure for these functions and employ ADOL‐C in an adapted gradient based optimisation method that was adjusted to the special properties of the objective functions considered here. First numerical results will be presented. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)